Skip to main content
×
Home

POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

  • MEHRNOOSH SADRZADEH (a1) and ROY DYCKHOFF (a2)
Abstract

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4, and S5, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.

Copyright
Corresponding author
*OXFORD UNIVERSITY COMPUTING LABORATORY, OXFORD, UK. E-mail:mehrs@comlab.ox.ac.uk
SCHOOL OF COMPUTER SCIENCE, ST ANDREWS UNIVERSITY, ST ANDREWS, SCOTLAND, UK. E-mail:rd@st-andrews.ac.uk
References
Hide All
Baltag A., Coecke B., & Sadrzadeh M. (2007). Epistemic actions as resources. Journal of Logic and Computation, 17, 555585.
Baltag A., & Moss L. (2004). Logics for epistemic programs. Synthese, 139, 165224.
Blackburn P., de Rijke M., & Venema Y. (2001). Modal Logic. Cambridge, UK: Cambridge University Press.
Bonnette N., & Goré R. (1998). A labelled sequent system for tense logic Kt. In Antoniou G., and Slaney J. K., editors. Australian Joint Conference on Artificial Intelligence, Volume 1502 of Lecture Notes in Computer Science. London: Springer, pp. 7182.
Brünnler K. (2006). Deep sequent systems for modal logic. In Governatori G., Hodkinson I., and Venema Y., editors. Advances in Modal Logic. Vol. 6. London: College Publications, pp. 107119.
Celani S., & Jansana R. (1999). Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic. Logic Journal of the IGPL, 7, 683715.
Dunn M. (2005). Positive modal logic. Studia Logica, 55, 301317.
Gehrke M., Nagahashi H., & Venema Y. (2005). A Sahlqvist theorem for distributive modal logic. Annals of Pure and Applied Logic, 131, 65102.
Goré R. (1998). Substructural logics on display. Logic Journal of the IGPL, 6(3), 451504.
Huth M., & Ryan M. (2000). Logic in Computer Science. Cambridge, UK: Cambridge University Press.
Kashima R. (1994). Cut-free sequent calculi for some tense logics. Studia Logica, 53, 119135.
Kriener J., Sadrzadeh M., & Dyckhoff R. (2009). Implementation of a cut-free sequent calculus for logics with adjoint modalities. Technical report, School of Computer Science, University of St Andrews, St Andrews, Scotland.
Moortgat M. (1995). Multimodal linguistic inference. Logic Journal of the IGPL, 3, 371401.
Negri S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34, 507544.
Prior A. (1968). Papers on Time and Tense. Oxford, UK: Oxford University Press.
Restall G. (2000). An Introduction to Substructural Logics. London: Routledge.
Richards S., & Sadrzadeh M. (2009). Aximo: Automated axiomatic reasoning for information update. Electronic Notes in Theoretical Computer Science, 231, 211225.
Sadrzadeh M. (2006). Actions and Resources in Epistemic Logic. PhD Thesis, Université du Québec à Montréal.
Sadrzadeh M. (2009). Ockham’s razor and reasoning about information flow. Synthese, 167, 391408.
Sadrzadeh M., & Dyckhoff R. (2009). Positive logic with adjoint modalities: Proof theory, semantics and reasoning about information. Electronic Notes in Theoretical Computer Science, 249, 451470.
Simpson A. (1993). The Proof Theory and Semantics of Intuitionistic Modal Logic. PhD Thesis, University of Edinburgh.
von Karger B. (1998). Temporal algebras. Mathematical Structures in Computer Science, 8, 277320.
Wansing H. (1994). Sequent calculi for normal modal propositional logics. Journal of Logic and Computation, 4(2), 125142.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 111 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th November 2017. This data will be updated every 24 hours.